Allan Variance Analysis of Measurement D
IMTC 2006 - Instrumentation and Measurement
Technology Conference
Warsaw, Poland, 1-3 May 2007
Allan Variance Analysis of Measurement Data Series for Instrument Verification
M. Bertoccol, A. Monetti2, E. Mottin1, C. Narduzzi1, E.
Sienil
'Department of Information Engineering, University of Padova,
via Gradenigo 6/b - 35131 Padova, Italy
Phone: +39 049 8277500, Fax: +39 049 8277699,
e-mail: matteo.bertocco, claudio.narduzzi, elisabetta.sieni@unipd.it
2Infineon Technologies Italia Srl, Development Center Padova,
via N. Tommaseo, 65/b - 35131 Padova, Italy.
Abstract - The paper is concerned with the analysis of measurand
and instrument stability during the calibration verification process,
with a view on ensuring that comparison is carried out under suitable
assumptions. It will be shown how analysis of the Allan variance
allows stability characterization of the measurement circuit and yields
information about measurement noise, so that a correct procedure
for the verification of the calibration status can be defined. This is
of particular significance in the context of an industrial laboratory,
where one of the main purposes of the analysis would be to identify
improvements to the measurement system, that could allow to perform
calibration verification within the limits of available equipment.
Keywords - Allan variance, measurement noise, calibration laboratory
I. INTRODUCTION
Quality control of production processes requires the periodic calibration of instrumentation used in tests and in process
measurements [1], [2]. Using an instrument in a non-valid calibration status invalidates measurements and would probably
require that they be repeated at the laboratory own cost.
Generally, instrument calibration is performed at suitably
competent laboratories, however this operation is rather expensive. To make the best use of resources while reducing the risk
of a non-valid calibration status, a laboratory may perform internal calibration verification in the time interval between two
calibrations, using procedures and instruments in accordance
with the indications given in [3]. In an industrial laboratory
environment the metrological support for this kind of activity
is generally far less comprehensive than in a dedicated calibration laboratory. This suggests the usefulness of investigating
some aspects of the verification process more closely, in the
hope of contributing some useful criteria to assure its correctness.
Verification of the calibration status of an instrument can
be performed by comparing measurement results obtained for
the same measurand using, respectively, a calibrated instrument acting as a working standard and the one under-test. The
1-4244-0589-0/07/$20.00 ©2007 IEEE
comparison criterion can be seen as a mathematical rule transforming measurement results into an index that, by comparison
with a threshold, allows a decision about the compatibility of
measurements. Most common criteria are based on statistical
indicators [4], [5], [6], that rely on hypotheses made about the
statistics of the data sets, such as their correlation, probability
distribution and uncertainty.
In common practice, data are supposed to be random, uncorrelated and, possibly, having a Gaussian probability distribution. Under these assumptions it can be considered that
accuracy in the estimation of the mean value and variance of
measurements improves as the number of values, N, in the
data set gets larger. However, if N is large, deviations from the
assumed conditions may occur, as a result of noise and trends
superposed on the measured values. These effects limit the
accuracy in the estimation of a quantity, since variance may
actually tend to diverge as the data set size N increases. Consequently, a more detailed analysis of the statistical properties
of the measurement data series with regards to time stability is
required.
The paper is mainly concerned with the analysis of measurand and instrument stability during the verification process,
with a view on ensuring that comparison is carried out under
suitable assumptions. It should be emphasized that the investigation into measurement fluctuations needs to account for the
whole measurement system [11], because contributions can be
due to the measurand as well as to instruments.
Two different but typical cases will be discussed: first, comparison of a digital multimeter with a calibrated one having
better resolution by an order of magnitude (6 1 versus 7 1 digits) will be considered. Then, comparison between multimeters
having similar performances (6 digits) will be analyzed: the rationale here is that, within a population of several similar multimeters, one is maintained as a working standard and kept under
stricter metrological control. In the former case, the aim is to
find out under which conditions a satisfactory test uncertainty
ratio (TUR) can be obtained; in the latter, conditions for carrying out suitable statistical tests must be determined. It will be
shown how analysis of the Allan variance allows stability characterization of the measurement circuit and yields information
about measurement noise, so that a correct procedure for the
verification of the calibration status can be defined. This is
of particular significance in the context of an industrial laboratory, where one of the main purposes of the analysis would
be to identify improvements to the measurement system, that
could allow to perform calibration verification within the limits
of available equipment.
II. ALLAN VARIANCE ANALYSIS
Measurement variability can be due to the variability of
the measurand itself, to variations in some influence quantity (e.g. environmental temperature) and, possibly, to noise
and drift within the measuring instrument itself. For instance,
Fig.1 shows a sequence of 5000 resistance measurements for
a standard commercial resistor with a nominal value of 10 Q,
obtained by a digital multimeter (Agilent 34401A). Measurements were taken in a room with no temperature control. It
is difficult to determine a mean value that can be a good representation of the measurand, as trends and local fluctuations
are rather clearly present in the time series of data. In other
words, these measurements should be dealt with, more appropriately, as a time series, whose relevant features need to be
analyzed. Of course, the assumption of a Gaussian distribution
would certainly not hold in this case for the whole data set.
Fluctuations are commonly described by a power-law spectral density model [14]:
S(f) =
Eh,f
(2)
ae
where a is an integer and it is assumed that -2 < a < +2. Although the model originates from the analysis of noise sources
in high stability frequency standards, it is in fact fairly general and has been extended to other kinds of standards (e.g.,
[10]). The term in (2) for which a = 0, represents white noise.
If its contribution dominates the model, a standard statistical
approach may be applicable. On the other hand, if the power
spectral density of the observed quantity is not white and values are correlated, use of standard methods to estimate Type A
uncertainty from acquired data, referred to in [7], is hardly justifiable [8], [9], [10]. For this purpose the Allan variance and
the power spectral density of the the series should be analyzed
[12], [13], [14].
The empirical plot of or (T) can be compared with the Allan variance model underlying (2), which is described by the
following expression [12]:
72(T)
=
J kltT.
(3)
/1
The coefficients kft are constants and u is related to a by:
j
=
{
-a 1
-2
for: a1
(4)
1.OE+O
a)
C)
1.OE-1
CZ
1.OE-2
Fig. 1. Set of 5000 measurement results for a 10 Q resistor (Agilent 34401A).
The Allan variance is commonly used in metrology to define the stability of a reference. Given a time series of data
indicated by y(kT), where T is the constant interval between
consecutive measurements, Allan variance is calculated by the
expression:
2(T)= I2N
koZ0+N-1 x
[E
k=ko
kzo+2N-1
)
S(
k=ko+N
-
x(kT)
2
(1)
with = NT and N a positive integer. Analysis of an empirical Allan variance plot can, in more general terms, provide
useful information about the stability of any kind of measurement system under analysis. Thus, it may prove a useful tool
in the analysis of the calibration verification process.
T
2
10
100
time [s]
Fig. 2. Time-variance plot for the Allan variance model (3) with:
k-I = 0.015, ko = 0.04, k+1 = 0.002.
Different values of ,u indicate different noise components:
=-1 refers to white noise, ,u = 0 refers to a 1/f noise
component, ,u = +1 to a random walk noise component and
= +2 to a deviation of the mean value, [10], [13]. Their relative weights, that depend on the constants kft, determine the
shape of the Allan variance plot.
Using plots of Allan variance versus averaging time, the
structure of the noise contribution can be evidenced and a maximum time interval in which noise can be considered white indicated, as discussed in [9]. Ideally, a U-shaped plot would result: this is shown in the example of Fig. 2, where a minimum
is obtained for an averaging time between 1 and 10 s. In this
,u
case further averaging would in no way improve the results.
In the following, some examples of the proposed use of Allan
variance plots in the analysis of the measurement procedure in
normal laboratory practice are shown.
III. PRACTICAL EXAMPLES
A. Verification based on TUR criteria
This subsection considers calibration verification of a 612
digit multimeter (Keithley 2000) by comparison with the indications of a 7 1 digit multimeter (Keithley 2001). Following
recommended practice [3], the value TUR = 4 is taken as a
reference.
Presented results refer to the DC voltage function: a voltage source is applied to the reference voltmeter and measured,
then the same source is measured by the multimeter under test
and readings compared. Results presented here refer to the
1 V range of the instrument under test, whose resolution is
1 ,uV; for the reference instrument the corresponding range is
2 V and the resolution 100 nV. It should be noticed that the
specified uncertainty of the K2000 multimeter in the 1 V range
is ±37,uV at full scale; therefore the specification TUR = 4
yields a requirement for a verification uncertainty not greater
than ±9,uV.
The reference multimeter is assumed to be fully calibrated,
therefore only Type A uncertainty will be accounted for. However this should make allowance for the measurement system stability, considering also the dead time between measurements taken with the two multimeters. Calibration verification
therefore requires the availability of a suitably stable voltage
source. This may be a critical point, in view of the resolution specifications of the instruments. Allan variance analysis
can then help assess whether calibration verification is feasible
with the measurement system at hand.
Figure 3 shows the Allan variance plots for measurements in
the 2 V range of the reference instrument. All plots were built
by acquiring a set of 8192 values; averaging involved at most
256 consecutive readings, so that each value is determined by
averaging at least 32 estimates. The curve represented by the
dashed line refers to the measurement of the output voltage of a
commercial adjustable laboratory power supply. It is useful to
compare it with that obtained by acquiring measurements with
a shorted input, represented by the continuous line. The former
accounts for instabilities and noise of the whole measurement
system, while the latter refers to the instrument alone.
Comparison between the two plots can provide an assessment of the relative importance of the instrument contribution
[15]. In this case the measurement system as a whole is characterized by a considerably larger Allan variance, about four
orders of magnitude higher. This can be attributed to voltage
fluctuations of the power supply output, while the continuous
line trace shows that intrinsic stability of the measuring instrument is much better.
3
Using the laboratory power supply as a reference does not
allow to reach the specified uncertainty target. Therefore, we
analyzed the feasibility of using a low-pass filter to stabilize
the power supply output. The first-order filter employed for
this purpose had a time constant of 560 s, resulting in a very
long settling time: it takes an estimated 2 1 hours for the filter
output to settle to within 0.1 ppm of the final value. Introduction of such unsophisticated, but very narrow-band low pass
filter achieves a surprising improvement: the curve marked by
'x' refers to measurements of the filtered power supply output
and practically coincides with the continuous line. This means
that, over time intervals up to 200 ms, the Allan deviation is of
the order of 100 nV, well below the target uncertainty.
10 8
10 10
C)- 1
:..
-.
...
.
60
Ii:
0#0.
::I
0
1 0 16
10 2
1o-1
100
integration time X [s]
101
102
Fig. 3. Allan variance plots for measurement of unfiltered voltage (dashed
line), filtered voltage ('x') and shorted input (continuous line), 2 V range
(Keithley 2001).
However, even after such a long time, only a relatively
short-term stability can be ensured for the generated voltage.
The upward slope in the right part of the plot is due to the slow
voltage variations related to the low-pass filter. These limit the
long-term stability of the measurement, putting an upper bound
of a few seconds on the time available to perform comparisons
with the instrument under test.
B. Data analysis for statistical tests
Results shown here are based on resistance measurement
data acquired using an Agilent 34401A multimeter. In this case
Allan variance analysis is mainly used to study noise contributions and their sources, as a preliminary step to the application
of statistical tests for compatibility of measurements. A set of
common commercial resistors mounted on a support provided
with banana plugs have been employed in a room without temperature control. Connections are provided by equal length
cables. The autozero function of the instrument has been disabled in order to obtain the maximum sampling rate, resulting
in a sampling interval of 200 ms. For each series of measurements 5000 samples have been taken; acquisitions are about 16
minutes long and are performed automatically using a PC and
a Lab VIEW program.
1.0118
xlO0
Dati
1.01161.01141.0112-
,1.011
U51.0108
1.0106
1.0104
1.0102
1 .01
1000
200
300
400
500
time (s)
600
700
800
900
1000
(a) time series.
10 .
10
..1 0.
101
101
10o
101
time [s]
10'
(b) Allan variance plot.
Fig. 4. Measurement results for a 1MQ resistor (Agilent 34401A).
Measured values for a 1 MQ resistor are shown in
Fig. 4(a), with the corresponding Allan variance being plotted
1 is followed by a flat tract
in Fig. 4(b). The initial slope u
evidencing, respectively, white and flicker noise contributions.
'The curve slope next shows a random walk noise contribution
(,u= + 1) and, finally, the effect of a trend in the measured
values becomes dominant (,u= +2).
An important aspect of the analysis is that the length of the
time window during which the assumption of a white noise
model is acceptable can be determined by evaluating the position of the first knee in the Allan variance curve, starting from
the slope of -1. For instance, the plot in Fig. 4(b) shows that,
for an acquisition time shorter than 2 s measurement variability
can be described as a white noise contribution superposed on
the measured values.
In this case the probability distribution of measured values might reasonably be assumed to be Gaussian, therefore
4
instrument comparison by statistical tests can rely on well established techniques. The window size corresponds to just 20
samples, but taking more values cannot improve measurement
accuracy [10]. On the other hand, this limitation also influences the confidence level that any statistical test on the compatibility of measurements can have.
For an acquisition window between 2 and 5 s the Allan variance remains constant, however a Gaussian distribution can no
longer be assumed. If the acquisition is longer than 5 s drift effects are evidenced. Therefore, with the measurement system
considered in this case the comparison of measurements obtained from two different instruments can be performed only if
acquisition time is shorter than 5 s, that means a comparatively
small data set. It should also be remembered that initial measurement conditions must be restored before the second set of
measurement acquisitions starts. Care has to be taken to ensure that conditions (e.g., resistor heating) in two subsequent
acquisitions can be assumed to remain reasonably constant.
Fig. 5 shows a comparison between the Allan variance plots
evaluated for two series of measurement values, obtained respectively with a 10 Q resistor and a short circuit connected
to the same instrument input. It can be noticed that the flat
tracts (,u= 0) associated with flicker noise coincide in the two
curves. This means that Type A measurement uncertainty is
bounded by the instrument contribution and corresponds to an
Allan variance of 2 x 10 ' Q2 . The final part of the plot for
the 10 Q resistor shows a linear trend with slope ,u= +2 that
highlights a drift in measured values, as seen in Fig. 1.
The curves in Fig. 5 also show a peculiar behavior for small
values of T, where a slope ,u= +1 is evidenced. This is associated with band limited white noise and is a consequence of
the instrument bandwidth limitation [ 14]. It should be noticed
that the effect is also dependent on the sampling frequency employed to acquire measurement data. It progressively disappears if measurements are taken with a longer sampling interval, which has the effect of decorrelating consecutive samples.
10-'
10
lo
short
10-6
107_
10 81
101
10o
101
time
[s]
102
103
Fig. 5. Comparison of Allan variances for resistance measurement by a
6-digit multimeter (Agilent 34401A).
This is practically demonstrated in Fig. 6, that shows the Allan
variance plots of the 10 Q resistor, calculated by varying the
sampling time between 0.2 s and 4 s. It can be seen that, by
increasing the sampling time, the initial slope of the curve is
reduced, while the level of flicker noise is not modified.
0.
....... ...............
::.
10v
-.-:..-.-:-.-:.
0.2
Tc =1 s
Tc = 2 s
Tc=4s
-
10-7_
10810-1
10
10
time [s]
102
103
Fig. 6. Evaluation of Allan variance for a 10 Q resistor, with different
sampling times.
IV. CONCLUSIONS
Allan variance results have been used to analyze the measurement systems used to verify the calibration status of digital
multimeters. In particular, resistance measurement data series
have been analyzed to decide about the maximum number of
samples that can be acquired to obtain uncorrelated realizations
of the measurand. It has been shown that the effects of changes
in the measurement system set-up can be studied by the analysis of Allan variance plots, evidencing different contributions.
The proposed approach can be used by test laboratories as a
help in the design of the procedures employed to verify the calibration status of their instrumentation. In particular the analysis can be used to determine better ways to reduce noise effects
or measurement trends and assess whether noise is primarily
due to the instrument or if the measurand noise contribution is
important.
REFERENCES
[1] ISO 9001, Quality management systems. Requirements, 2000.
[2] ISO/IEC 17025, General requirements for the competence of testing and
calibration laboratories, 1999.
[3] UNI EN ISO 10012, Measurement management systems. Requirements
for measurement processes and measuring equipment, 2004, (Interantional reference: ISO 10012: 2003).
[4] G. Zingales, "Compatibility in industrial measurements," Proc. ofIMTC,
pp. 377-380, 1999.
[5] E. Arri, F. Cabiati, S. D'Emilio, and L. Gonella, "On the application of
the expression of uncertainty in measurement to measuring instruments,"
Measurement, pp. 51-57, 1995.
5
[6] EA-2-03, (EAL-P7), EAL Interlaboratory Comparisons, 1996.
[7] ENV 13005, Guide to the expression of uncertainty in measurement,
1999.
[8] T.J. Witt, "Low-frequency spectral analysis of DCnanovoltmeters and
voltage reference standards," IEEE Trans. on Instrument.n and Meas.,
vol. 46, pp. 318-321, 1997.
[9] P. Helist6 and H. Seppa, "Measurement uncertainty in the presence of
low-frequency noise," IEEE Trans. on Instrument.n and Meas., vol. 50,
pp. 453-456, 2001.
[10] T.J. Witt and D. Reymann, "Using power spectra and Allan variance to
characterise the noise of Zener-diode voltage standards," IEEE Proc-Sci
Meas. Technol., vol. 147, pp. 177-182, 2000.
[11] J. Valdes, M. E. Porfiri, E. E. Lobbe, F. Kornblit, M. N. Passarino
de Marques, and J. A. Leiblich, "Long term fluctuations associated with
different standards," IEEE Trans. on Instrument.n and Meas., vol. 42,
pp. 269-272, 1993.
[12] D. W. Allan, "Should the classical variance be used as a basic measure
in standards metrology?," IEEE Trans. on Instrument.n and Meas., vol.
147, pp. 646-654, 1987.
[13] T.J. Witt, "Using the Allan variance and power spectral density to characterize DC nanovoltmeters," IEEE Trans. on Instrument.n and Meas.,
vol. 50, pp. 445-448, 2001.
[14] J. Rutman and F.L. Walls, "Characterization of frequency stability in
precision frequency sources," Proc.s of the IEEE, vol. 79, pp. 952-960,
1991.
[15] T.J. Witt, "Allan variances and spectral densities for DC voltage measurements with polarity reversals," IEEE Trans. on Instrument.n and
Meas., vol. 54, pp. 550 - 553, 2005.
[16] Agilent Technologies, Agilent 34401A, Service's Guide, 1996.
Technology Conference
Warsaw, Poland, 1-3 May 2007
Allan Variance Analysis of Measurement Data Series for Instrument Verification
M. Bertoccol, A. Monetti2, E. Mottin1, C. Narduzzi1, E.
Sienil
'Department of Information Engineering, University of Padova,
via Gradenigo 6/b - 35131 Padova, Italy
Phone: +39 049 8277500, Fax: +39 049 8277699,
e-mail: matteo.bertocco, claudio.narduzzi, elisabetta.sieni@unipd.it
2Infineon Technologies Italia Srl, Development Center Padova,
via N. Tommaseo, 65/b - 35131 Padova, Italy.
Abstract - The paper is concerned with the analysis of measurand
and instrument stability during the calibration verification process,
with a view on ensuring that comparison is carried out under suitable
assumptions. It will be shown how analysis of the Allan variance
allows stability characterization of the measurement circuit and yields
information about measurement noise, so that a correct procedure
for the verification of the calibration status can be defined. This is
of particular significance in the context of an industrial laboratory,
where one of the main purposes of the analysis would be to identify
improvements to the measurement system, that could allow to perform
calibration verification within the limits of available equipment.
Keywords - Allan variance, measurement noise, calibration laboratory
I. INTRODUCTION
Quality control of production processes requires the periodic calibration of instrumentation used in tests and in process
measurements [1], [2]. Using an instrument in a non-valid calibration status invalidates measurements and would probably
require that they be repeated at the laboratory own cost.
Generally, instrument calibration is performed at suitably
competent laboratories, however this operation is rather expensive. To make the best use of resources while reducing the risk
of a non-valid calibration status, a laboratory may perform internal calibration verification in the time interval between two
calibrations, using procedures and instruments in accordance
with the indications given in [3]. In an industrial laboratory
environment the metrological support for this kind of activity
is generally far less comprehensive than in a dedicated calibration laboratory. This suggests the usefulness of investigating
some aspects of the verification process more closely, in the
hope of contributing some useful criteria to assure its correctness.
Verification of the calibration status of an instrument can
be performed by comparing measurement results obtained for
the same measurand using, respectively, a calibrated instrument acting as a working standard and the one under-test. The
1-4244-0589-0/07/$20.00 ©2007 IEEE
comparison criterion can be seen as a mathematical rule transforming measurement results into an index that, by comparison
with a threshold, allows a decision about the compatibility of
measurements. Most common criteria are based on statistical
indicators [4], [5], [6], that rely on hypotheses made about the
statistics of the data sets, such as their correlation, probability
distribution and uncertainty.
In common practice, data are supposed to be random, uncorrelated and, possibly, having a Gaussian probability distribution. Under these assumptions it can be considered that
accuracy in the estimation of the mean value and variance of
measurements improves as the number of values, N, in the
data set gets larger. However, if N is large, deviations from the
assumed conditions may occur, as a result of noise and trends
superposed on the measured values. These effects limit the
accuracy in the estimation of a quantity, since variance may
actually tend to diverge as the data set size N increases. Consequently, a more detailed analysis of the statistical properties
of the measurement data series with regards to time stability is
required.
The paper is mainly concerned with the analysis of measurand and instrument stability during the verification process,
with a view on ensuring that comparison is carried out under
suitable assumptions. It should be emphasized that the investigation into measurement fluctuations needs to account for the
whole measurement system [11], because contributions can be
due to the measurand as well as to instruments.
Two different but typical cases will be discussed: first, comparison of a digital multimeter with a calibrated one having
better resolution by an order of magnitude (6 1 versus 7 1 digits) will be considered. Then, comparison between multimeters
having similar performances (6 digits) will be analyzed: the rationale here is that, within a population of several similar multimeters, one is maintained as a working standard and kept under
stricter metrological control. In the former case, the aim is to
find out under which conditions a satisfactory test uncertainty
ratio (TUR) can be obtained; in the latter, conditions for carrying out suitable statistical tests must be determined. It will be
shown how analysis of the Allan variance allows stability characterization of the measurement circuit and yields information
about measurement noise, so that a correct procedure for the
verification of the calibration status can be defined. This is
of particular significance in the context of an industrial laboratory, where one of the main purposes of the analysis would
be to identify improvements to the measurement system, that
could allow to perform calibration verification within the limits
of available equipment.
II. ALLAN VARIANCE ANALYSIS
Measurement variability can be due to the variability of
the measurand itself, to variations in some influence quantity (e.g. environmental temperature) and, possibly, to noise
and drift within the measuring instrument itself. For instance,
Fig.1 shows a sequence of 5000 resistance measurements for
a standard commercial resistor with a nominal value of 10 Q,
obtained by a digital multimeter (Agilent 34401A). Measurements were taken in a room with no temperature control. It
is difficult to determine a mean value that can be a good representation of the measurand, as trends and local fluctuations
are rather clearly present in the time series of data. In other
words, these measurements should be dealt with, more appropriately, as a time series, whose relevant features need to be
analyzed. Of course, the assumption of a Gaussian distribution
would certainly not hold in this case for the whole data set.
Fluctuations are commonly described by a power-law spectral density model [14]:
S(f) =
Eh,f
(2)
ae
where a is an integer and it is assumed that -2 < a < +2. Although the model originates from the analysis of noise sources
in high stability frequency standards, it is in fact fairly general and has been extended to other kinds of standards (e.g.,
[10]). The term in (2) for which a = 0, represents white noise.
If its contribution dominates the model, a standard statistical
approach may be applicable. On the other hand, if the power
spectral density of the observed quantity is not white and values are correlated, use of standard methods to estimate Type A
uncertainty from acquired data, referred to in [7], is hardly justifiable [8], [9], [10]. For this purpose the Allan variance and
the power spectral density of the the series should be analyzed
[12], [13], [14].
The empirical plot of or (T) can be compared with the Allan variance model underlying (2), which is described by the
following expression [12]:
72(T)
=
J kltT.
(3)
/1
The coefficients kft are constants and u is related to a by:
j
=
{
-a 1
-2
for: a1
(4)
1.OE+O
a)
C)
1.OE-1
CZ
1.OE-2
Fig. 1. Set of 5000 measurement results for a 10 Q resistor (Agilent 34401A).
The Allan variance is commonly used in metrology to define the stability of a reference. Given a time series of data
indicated by y(kT), where T is the constant interval between
consecutive measurements, Allan variance is calculated by the
expression:
2(T)= I2N
koZ0+N-1 x
[E
k=ko
kzo+2N-1
)
S(
k=ko+N
-
x(kT)
2
(1)
with = NT and N a positive integer. Analysis of an empirical Allan variance plot can, in more general terms, provide
useful information about the stability of any kind of measurement system under analysis. Thus, it may prove a useful tool
in the analysis of the calibration verification process.
T
2
10
100
time [s]
Fig. 2. Time-variance plot for the Allan variance model (3) with:
k-I = 0.015, ko = 0.04, k+1 = 0.002.
Different values of ,u indicate different noise components:
=-1 refers to white noise, ,u = 0 refers to a 1/f noise
component, ,u = +1 to a random walk noise component and
= +2 to a deviation of the mean value, [10], [13]. Their relative weights, that depend on the constants kft, determine the
shape of the Allan variance plot.
Using plots of Allan variance versus averaging time, the
structure of the noise contribution can be evidenced and a maximum time interval in which noise can be considered white indicated, as discussed in [9]. Ideally, a U-shaped plot would result: this is shown in the example of Fig. 2, where a minimum
is obtained for an averaging time between 1 and 10 s. In this
,u
case further averaging would in no way improve the results.
In the following, some examples of the proposed use of Allan
variance plots in the analysis of the measurement procedure in
normal laboratory practice are shown.
III. PRACTICAL EXAMPLES
A. Verification based on TUR criteria
This subsection considers calibration verification of a 612
digit multimeter (Keithley 2000) by comparison with the indications of a 7 1 digit multimeter (Keithley 2001). Following
recommended practice [3], the value TUR = 4 is taken as a
reference.
Presented results refer to the DC voltage function: a voltage source is applied to the reference voltmeter and measured,
then the same source is measured by the multimeter under test
and readings compared. Results presented here refer to the
1 V range of the instrument under test, whose resolution is
1 ,uV; for the reference instrument the corresponding range is
2 V and the resolution 100 nV. It should be noticed that the
specified uncertainty of the K2000 multimeter in the 1 V range
is ±37,uV at full scale; therefore the specification TUR = 4
yields a requirement for a verification uncertainty not greater
than ±9,uV.
The reference multimeter is assumed to be fully calibrated,
therefore only Type A uncertainty will be accounted for. However this should make allowance for the measurement system stability, considering also the dead time between measurements taken with the two multimeters. Calibration verification
therefore requires the availability of a suitably stable voltage
source. This may be a critical point, in view of the resolution specifications of the instruments. Allan variance analysis
can then help assess whether calibration verification is feasible
with the measurement system at hand.
Figure 3 shows the Allan variance plots for measurements in
the 2 V range of the reference instrument. All plots were built
by acquiring a set of 8192 values; averaging involved at most
256 consecutive readings, so that each value is determined by
averaging at least 32 estimates. The curve represented by the
dashed line refers to the measurement of the output voltage of a
commercial adjustable laboratory power supply. It is useful to
compare it with that obtained by acquiring measurements with
a shorted input, represented by the continuous line. The former
accounts for instabilities and noise of the whole measurement
system, while the latter refers to the instrument alone.
Comparison between the two plots can provide an assessment of the relative importance of the instrument contribution
[15]. In this case the measurement system as a whole is characterized by a considerably larger Allan variance, about four
orders of magnitude higher. This can be attributed to voltage
fluctuations of the power supply output, while the continuous
line trace shows that intrinsic stability of the measuring instrument is much better.
3
Using the laboratory power supply as a reference does not
allow to reach the specified uncertainty target. Therefore, we
analyzed the feasibility of using a low-pass filter to stabilize
the power supply output. The first-order filter employed for
this purpose had a time constant of 560 s, resulting in a very
long settling time: it takes an estimated 2 1 hours for the filter
output to settle to within 0.1 ppm of the final value. Introduction of such unsophisticated, but very narrow-band low pass
filter achieves a surprising improvement: the curve marked by
'x' refers to measurements of the filtered power supply output
and practically coincides with the continuous line. This means
that, over time intervals up to 200 ms, the Allan deviation is of
the order of 100 nV, well below the target uncertainty.
10 8
10 10
C)- 1
:..
-.
...
.
60
Ii:
0#0.
::I
0
1 0 16
10 2
1o-1
100
integration time X [s]
101
102
Fig. 3. Allan variance plots for measurement of unfiltered voltage (dashed
line), filtered voltage ('x') and shorted input (continuous line), 2 V range
(Keithley 2001).
However, even after such a long time, only a relatively
short-term stability can be ensured for the generated voltage.
The upward slope in the right part of the plot is due to the slow
voltage variations related to the low-pass filter. These limit the
long-term stability of the measurement, putting an upper bound
of a few seconds on the time available to perform comparisons
with the instrument under test.
B. Data analysis for statistical tests
Results shown here are based on resistance measurement
data acquired using an Agilent 34401A multimeter. In this case
Allan variance analysis is mainly used to study noise contributions and their sources, as a preliminary step to the application
of statistical tests for compatibility of measurements. A set of
common commercial resistors mounted on a support provided
with banana plugs have been employed in a room without temperature control. Connections are provided by equal length
cables. The autozero function of the instrument has been disabled in order to obtain the maximum sampling rate, resulting
in a sampling interval of 200 ms. For each series of measurements 5000 samples have been taken; acquisitions are about 16
minutes long and are performed automatically using a PC and
a Lab VIEW program.
1.0118
xlO0
Dati
1.01161.01141.0112-
,1.011
U51.0108
1.0106
1.0104
1.0102
1 .01
1000
200
300
400
500
time (s)
600
700
800
900
1000
(a) time series.
10 .
10
..1 0.
101
101
10o
101
time [s]
10'
(b) Allan variance plot.
Fig. 4. Measurement results for a 1MQ resistor (Agilent 34401A).
Measured values for a 1 MQ resistor are shown in
Fig. 4(a), with the corresponding Allan variance being plotted
1 is followed by a flat tract
in Fig. 4(b). The initial slope u
evidencing, respectively, white and flicker noise contributions.
'The curve slope next shows a random walk noise contribution
(,u= + 1) and, finally, the effect of a trend in the measured
values becomes dominant (,u= +2).
An important aspect of the analysis is that the length of the
time window during which the assumption of a white noise
model is acceptable can be determined by evaluating the position of the first knee in the Allan variance curve, starting from
the slope of -1. For instance, the plot in Fig. 4(b) shows that,
for an acquisition time shorter than 2 s measurement variability
can be described as a white noise contribution superposed on
the measured values.
In this case the probability distribution of measured values might reasonably be assumed to be Gaussian, therefore
4
instrument comparison by statistical tests can rely on well established techniques. The window size corresponds to just 20
samples, but taking more values cannot improve measurement
accuracy [10]. On the other hand, this limitation also influences the confidence level that any statistical test on the compatibility of measurements can have.
For an acquisition window between 2 and 5 s the Allan variance remains constant, however a Gaussian distribution can no
longer be assumed. If the acquisition is longer than 5 s drift effects are evidenced. Therefore, with the measurement system
considered in this case the comparison of measurements obtained from two different instruments can be performed only if
acquisition time is shorter than 5 s, that means a comparatively
small data set. It should also be remembered that initial measurement conditions must be restored before the second set of
measurement acquisitions starts. Care has to be taken to ensure that conditions (e.g., resistor heating) in two subsequent
acquisitions can be assumed to remain reasonably constant.
Fig. 5 shows a comparison between the Allan variance plots
evaluated for two series of measurement values, obtained respectively with a 10 Q resistor and a short circuit connected
to the same instrument input. It can be noticed that the flat
tracts (,u= 0) associated with flicker noise coincide in the two
curves. This means that Type A measurement uncertainty is
bounded by the instrument contribution and corresponds to an
Allan variance of 2 x 10 ' Q2 . The final part of the plot for
the 10 Q resistor shows a linear trend with slope ,u= +2 that
highlights a drift in measured values, as seen in Fig. 1.
The curves in Fig. 5 also show a peculiar behavior for small
values of T, where a slope ,u= +1 is evidenced. This is associated with band limited white noise and is a consequence of
the instrument bandwidth limitation [ 14]. It should be noticed
that the effect is also dependent on the sampling frequency employed to acquire measurement data. It progressively disappears if measurements are taken with a longer sampling interval, which has the effect of decorrelating consecutive samples.
10-'
10
lo
short
10-6
107_
10 81
101
10o
101
time
[s]
102
103
Fig. 5. Comparison of Allan variances for resistance measurement by a
6-digit multimeter (Agilent 34401A).
This is practically demonstrated in Fig. 6, that shows the Allan
variance plots of the 10 Q resistor, calculated by varying the
sampling time between 0.2 s and 4 s. It can be seen that, by
increasing the sampling time, the initial slope of the curve is
reduced, while the level of flicker noise is not modified.
0.
....... ...............
::.
10v
-.-:..-.-:-.-:.
0.2
Tc =1 s
Tc = 2 s
Tc=4s
-
10-7_
10810-1
10
10
time [s]
102
103
Fig. 6. Evaluation of Allan variance for a 10 Q resistor, with different
sampling times.
IV. CONCLUSIONS
Allan variance results have been used to analyze the measurement systems used to verify the calibration status of digital
multimeters. In particular, resistance measurement data series
have been analyzed to decide about the maximum number of
samples that can be acquired to obtain uncorrelated realizations
of the measurand. It has been shown that the effects of changes
in the measurement system set-up can be studied by the analysis of Allan variance plots, evidencing different contributions.
The proposed approach can be used by test laboratories as a
help in the design of the procedures employed to verify the calibration status of their instrumentation. In particular the analysis can be used to determine better ways to reduce noise effects
or measurement trends and assess whether noise is primarily
due to the instrument or if the measurand noise contribution is
important.
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5
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